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Fall 2012: Project 3: Bridge

Objective:
Construct a bridge that has the largest load to bridge weight ratio.

Materials:

Corrugated Cardboard

Glue

Rules:

1.The bridge must
span a 4 foot gap.

2.The bridge can be
no more than 6 inches wide and 3 inches below the height of the testing
surface.

3.There is no limit
to it being longer than 4 feet and there is no limit to its height above the
testing surface.

4.The bridge will
be loaded with a rope attached to a metal plate that must sit on the bed of the
bridge. The bridge must have a hole 1 inch in diameter and a flat surface of 3
x 3 inches to support the plate. The plate will be placed on the bridge at the
center of the 4 foot gap and at the height of the test surface

5.Calculation of
load to bridge weight ratio:

R = (weight supported by bridge before collapsing)

(weight of finished bridge)

Side-view of Testing Rules

3rd Place Winner

2nd Place Winner

1st Place Winner

Truss Analysis

Building bridges can be very complicated, but one of the
simplest designs to build and analyze is the truss bridge. The truss bridge
consists of straight members (ie., steel plates) and joints (ie., rivets). Usually,
like the bridge above, the members are jointed in a vertical plane to form a
repeating pattern. The members and joints can be jointed into very complex
patterns, but regardless of the pattern, truss bridges can be analyzed using the
same method. The analysis takes advantage of the fact that each member is in
tension or compression, and this force acts along the length of the member.
This means that each member either pulls or pushes on the joints it is attached
to and this force lies on a line going through the member. Also, the entire
bridge is in static equilibrium; the net forces in all directions and the net
torques about all points in space are zero. The bridge is not rotating or
moving.

Steps:

1. Create
a drawing of the members and joints. Label the angles, lengths of the
members, and external loads. Here is an example of one side of a truss
bridge made from three isosceles triangles, with a load applied in the
center and supports on either end.

2. Determine
the external forces. We already know the external load. There is a
support on either end with unknown forces. Since the bridge is in static
equilibrium, the net torque about any point must equal zero and the net
forces in any direction must also equal zero. Assume the members have no
mass, and solve for F1 and F2 by first solving for
the forces in the vertical direction and then taking the torques about point
A.

3. Determine the internal forces. In order to solve for the internal forces, we will
use the cutting method. Image cutting the bridge in half along the joints and
saving only one of the halves. The members we cut through had internal forces
in them, but we did not draw them because the force of the member on the joint
was balanced by an equal and opposite force of the joint on the member.
Although the member has been imaginatively removed, the force of the member on
the joint is still there and must be accounted for, so we will draw this force
in our diagram. Here we removed the left half of the bridge, and placed unknown
forces at points A and B in the direction of the members that would have
connected there. The members are arbitrarily assumed to be in tension, so F1
and F2 point away from the joints. If they are really in
compression, we will get a negative value for F1 or F2.
Again, since the bridge is in static equilibrium, the net torque about any
point must equal zero and the net forces in any direction must also equal zero.
Assume the members have no mass, and solve for F1 and F2
by first solving for the torques about point A and then about point B.

F1 represents the force of the member
on this joint. Since F1 is negative, it really points the opposite
direction to the one drawn. The force of the joint on the member will be equal and
opposite to F1. So, F1 pushes on the joint while joint A pushes on the member,
meaning the member is in compression.4. Repeat
Step 3 as many times as are needed. Cut the bridge at different
locations, so the forces in different members may be “exposed” and solved
for. When completed, it should look like this:

From this analysis, the members on the outside are in
compression while the members on the inside are in tension. Since we are
working with cardboard, there is a greater chance that the cardboard will buckle
under compression than rip under tension. Members 1, 2, and 5 should be made
thicker and with “fins” to prevent buckling.

Calculating the forces can be complicated, so here is a website by Johns Hopkins University which allows you to draw and automatically analyze the truss bridge: Virtual Laboratory Bridge Designer.